Page 370 - Petrophysics 2E

338 PETROPHYSICS: RESERVOIR ROCK PROPERTIES

the inlet capillary pressure, dS/d(P,) must be evaluated from the
experimental data. This presents a difficult problem because the data
have inherent errors that produce large errors in the derivatives. The
various approaches for the analyses of centrifuge capillary pressure
measurements differ in the way that the derivative is evaluated.
Donaldson et al. found that a least-square solution of a hyperbolic
function represented all capillary pressure curves from the literature
that were examined, as well as curves obtained from samples that were
treated to establish extremes of water-wet and oil-wet conditions [20].
Using the experimental data, the constants A, B, and C are evaluated and
then the derivative required by Equation 5.43 is evaluated:

(5.44)

Differentiating Equation 5.44:

d(Pc)i = [ B+AC ] x dS (5.45)

(1 + CS)2

Substituting into Equation 5.43:

(1 + csy
Si = S + (Pc)i x [ B-AC ] (5.46)

Using this method, the noise of experimental errors is removed by
the least-squares fit of the experimental data using Equation 5.44. Thus,
the saturation at the inlet face of the core, subject to the Hassler and
Brunner assumption, may be readily calculated from Equation 5.46. This
saturation corresponds to the capillary pressure at the inlet face of the
core, calculated using Equation 5.31. The details of this procedure are
presented in the example on page 327.

THEORETICALLY EXACT CALCULATION OF THE INLET SATURATION

Several attempts have been made to obtain an exact method for
calculating the inlet-face saturation. Hassler and Brunner proposed a
procedure that involves successive iterations to solve the basic equation
without making the simplifying assumptions, but these iterations
introduce approximations [ 171. Van Domselaar showed the derivation
of the basic equation beginning with Equation 5.38 and replaced the